A closer look at the implication

It seems necessary to further investigate the 4 possible situations of the example ‘if it rains then the picnic will canceled’. I’ll explain it in propositional logic.

1st, the implication for this statement has 2×2 values, representing all possible situations; 2 for rain and 2 for  canceled picnic. The 2 variables (A and B, representing rain and cancelled picnic) are placed in 2 columns to compare their possible values. The implication symbol, the arrow, calculates both the variable’s (truth) values. Those values can be expressed in 1 and 0 or True and False.

In logic, if a statement is true, then its negation (~) is false. If a statement is false, then its negation is true.
If A=T then ~A=F.
If A=F then ~A=T.

implication

value

meaning

A

B

formulas

If it rains, then the picnic will be canceled

T

T

T

A→B

rain → cancelled picnic

T

F

F

~(A→~B)

It is not the case that if it rains → picnic not cancelled

F

T

T

~(B→~A)

It is not the case that if picnic is cancelled then no rain (picnic cancelled, despite no rain) (=possible)

F

T

F

~B→~A

no cancelled picnic → no rain

Notice that A→B logically means exactly the same as ~B→~A the difference is that A→B is stated affirmative and ~B→~A is stated negative. ~B→~A is called the contrapositive of A→B, which is a direct statement.

Although A=T and B=F in ~(A→~B), since the whole statement is negated, its outcome is denied.

Although B=T and A=F in ~(B→~A), since the whole statement is also negated, there is a possibility. The not-raining is detached as sole cause for the cancelled picnic. Since this valid possibility of the implication is not an intuitive formula, it is easily overlooked. Hence, the thought that if the antecedent is denied, that thus follows the consequent has to be denied as well.

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